Life is invariably fatal: 100% of people die. Or do they? It is frequently claimed, by people who do not pause to think about what they are saying, that more people are alive today than have ever died. That’s nonsense. There are currently, according to UN estimates, just over 7.7 billion people on Earth. Lack of accurate data, or indeed any data at all for much of the time, makes it difficult even to guess the number of people in past times, but the table gives plausible estimates for the years in which landmark figures were reached.

From these figures, we may estimate that in the 500 years between 600 and 1100, well over a billion people (5 200 million) died, as very few people lived more than a hundred years. Another billion deaths would have been exceeded between 1100 and 1400, followed by more than 2 billion from 1400 to 1800. When we add the billion alive in 1804 and the 2 billion in 1927, we are already well over 7 billion.

Homo sapiens emerged between 50,000 and 300,000 years so, which adds a large number of dead people to our collection. We can only guess at population sizes and average lifespans in the early years but the best-informed estimates have reached the conclusion that around 108 billion people have ever lived. This means that about one-fourteenth of the people who have ever lived are alive today.

Looking on the bright side, this means that only 13 out of every 14 people who have ever been born have died so we might optimistically conclude that we have a 1 in 14 chance of living forever. Right?

Er, no. Just wait another 100 years or so and we’ll have the complete data on almost all of us. And that introduces the real problem of estimating life expectancy.

Having complete data is a problem often ignored by overeager users of medical statistics. What should we make, for example, of a recent report that deaths from breast cancer have been going down by almost 2% per year? Since everyone dies, a reduction in deaths from one cause must be matched by an increase in others. A decrease in premature deaths means something; an overall decrease needs further probing. And what does it mean when we read that life expectancy at birth in the UK is 79.6 years? It certainly doesn’t mean that the average age at which British people die is 79.6, because anyone who is 79 years old was born 79 years ago, not today. To predict how long a person born today will live involves making a prediction of medical progress for the next century. The spurious accuracy of that 79.6 figure covers a calculation based on a variety of assumptions, many of which are not easy to justify. Yet published figures of life expectancy have a huge economic effect on pension funds and government planning. Before considering the present state of life expectancy calculations, however, let us go back to its beginnings.

The earliest known collection and publication of data in a form that resembled a life expectancy table was by the Roman jurist Ulpian around AD 220. Much admired at the time as a legal authority, he advised on a system of taxation and inheritance payments that involved a death tax of around 5% on any legacy, with the remaining 95% funding an annual payment to the recipient of the legacy at a prescribed rate in a manner similar to modern annuities. To determine a fair rate, however, an estimate was needed of the life expectancy of the recipient and that was what Ulpian’s figures set out to provide.

Where his figures came from is not known, and great doubts have been expressed concerning their reliability and the statistical methods used for their calculation, but they suggest a female life expectancy at birth of 22.5 years and a male life expectancy of 20.4. For anyone reaching their late 30s, however, Ulpian predicts another 20 years of life while the over-60s could count on another five years on average. Ulpian himself lived to his early 50s. He was murdered in AD 223 in a riot between the soldiers and the mob by members of the Praetorian Guard, whom he had annoyed by reducing their privileges some years earlier.

Ulpian’s tables remained the last word in life expectancy predictions in the Roman Empire for several hundred years and were not surpassed significantly until the seventeenth century when an Englishman, whose name is now associated with a less earthly, more celestial observation, had a very bright idea. That man was to become England’s second Astronomer Royal and his name was Edmond (sometimes spelt Edmund) Halley.

Quite apart from his discovery of the comet named after him and his correct prediction of its return in a 76-year cycle, Halley made prodigious contributions to a number of scientific fields from an early age. He went to Queen’s College, Oxford, at the age of 16 and published papers on sunspots and the Solar System while still an undergraduate. He left Oxford after four years, without having taken a degree, to set up an observatory on the island of Saint Helena. Oxford made up for his formal failure to graduate by giving him an MA degree when he was 22, at which age he was also elected to be a Fellow of the Royal Society. He died at the age of 86, in 1742, allegedly after drinking a glass of wine against his doctor’s orders.

Halley’s great contribution to the science of life expectancy came in 1693 when he had been working in the Austrian town of Breslau (now Wrocław in Poland). He had come across data recording the annual numbers of births and deaths in the town over a five-year period and, most importantly, the sex and age of those who had died. Since Breslau was a small, tightly knit community far from the sea, and the number of births was roughly equal to the number of deaths over the five-year period, Halley reasoned that the number of people joining or leaving the town each year, other than through births and deaths, was small. This meant the total population (for which the records gave no precise figure) was reasonably stable, and that assumption enabled Halley to draw far-reaching conclusions.

His method was simple: the assumption of a constant birth rate told him the number of people of any age who would be alive if none of them had died, and by summing the mortality numbers for all lower ages, he calculated how many were still alive. This technique enabled him to calculate the odds an individual of any specified age had of reaching his next birthday. His table, for example, gave the number of 25-year-olds living in Breslau as 567 while the number of 26-year-olds was 560, so in his own words: ‘As for Instance, a Person of 25 Years of Age has the odds of 560 to 7 or 80 to 1 that he does not die in a year: Because that of 567, living 25 years of Age there die no more than 7 in a year, leaving 560 of 26 Years old.’

Halley did not specifically calculate life expectancy at birth, but we can see from his figures that around 50% of people died before they were 34. He did, however, devote a good deal of space to the calculation of sensible rates of annuities, as he made clear in the full title of his report: ‘An estimate of the degrees of the mortality of mankind; drawn from curious tables of the births and funerals at the city of Breslaw; with an attempt to ascertain the price of annuities upon lives.’

In fact, much of the motivation behind Halley’s work in this respect lay in the English government’s policy of raising funds for the war against France by selling annuities but the rates they offered were considerably less justifiable than those of Ulpian back in ancient Rome.

At one stage, the English government offered annuities that paid back their full price in only seven years; even when that time frame was doubled to 14 years, no account was made for the age of the purchaser. Nowadays, annuity rates are based on mortality tables. The total amount invested is usually built up over an individual’s working life, then used to purchase the annuity, which guarantees to pay a certain amount each year. The word ‘annuity’ comes from annus, the Latin for ‘year’, and the annual amount depends on the age of the individual and the number of further years they can expect to live. Back in the seventeenth century, the economist William Petty and the statistician John Graunt had made valiant efforts to draw up mortality tables for London some 30 years before Halley came along, but they lacked the precise data to calculate life expectancy that the town of Breslau offered, so their figures offered only limited help in making financial decisions. Without their contributions, however, Halley would probably not have had the idea of using the data in the way he did.

Today, it is clear that statisticians still rely on the basic techniques introduced by Halley, though the pace of change in modern life requires some significant changes and additions to those techniques.

When the Office for National Statistics tells us that life expectancy at birth in the UK is 87.6 years for a man and 90.2 years for a woman, what exactly does that mean and how did they calculate it?